chapter 11 ALGEBRAIC SYSTEMS GOALS The primary goal of this chapter is to make the reader aware of what an algebraic system is and how algebraic systems can be studied at different levels of abstraction. After describing the concrete, axiomatic, and universal levels, we will introduce one of the most important algebraic systems at the axiomatic level, the group. In this chapter, group theory will be a vehicle for introducing the universal concepts of isomorphism, direct product, subsystem, and generating set. These concepts can be applied to all algebraic systems. The simplicity of group theory will help the reader obtain a good intuitive understanding of these concepts. In Chapter 15, we will introduce some additional concepts and applications of group theory. We will close the chapter with a discussion of how some computer hardware and software systems use the concept of an algebraic system. Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommerical-ShareAlike 3.0 United States License. Chapter 11 - Algebraic Systems 11.1 Operations One of the first mathematical skills that we all learn is how to add a pair of positive integers. A young child soon recognizes that something is wrong if a sum has two values, particularly if his or her sum is different from the teacher's. In addition, it is unlikely that a child would consider assigning a non-positive value to the sum of two positive integers. In other words, at an early age we probably know that the sum of two positive integers is unique and belongs to the set of positive integers. This is what characterizes all binary operations on a set. Definition: Binary Operation. Let S be a nonempty set. A binary operation on S is a rule that assigns to each ordered pair of elements of S a unique element of S. In other words, a binary operation is a function from Sµ S into S. Example 11.1.1. Union and intersection are both binary operations on the power set of any universe. Addition and multiplication are binary operators on the natural numbers. Addition and multiplication are binary operations on the set of 2 by 2 real matrices, M2µ2HRL. Division is a binary operation on some sets of numbers, such as the positive reals. But on the integers (1ê2 – Z) and even on the real numbers H1ê0 is not defined), division is not a binary operation. Notes: (a) We stress that the image of each ordered pair must be in S. This requirement disqualifies subtraction on the natural numbers from consideration as a binary operation, since 1 - 2 is not a natural number. Subtraction is a binary operation on the integers. (b) On Notation. Despite the fact that a binary operation is a function, symbols, not letters, are used to name them. The most commonly used symbol for a binary operation is an asterisk, *. We will also use a diamond, ù, when a second symbol is needed. (c) If * is a binary operation on S and a, b œ S, there are three common ways of denoting the image of the pair (a, b). They are: *a b a*b a b * Prefix Form Infix Form Postfix FOrm We are all familiar with infix form. For example, 2 + 3 is how everyone is taught to write the sum of 2 and 3. But notice how 2 + 3 was just described in the previous sentence! The word sum preceded 2 and 3. Orally, prefix form is quite natural to us. The prefix and postfix forms are superior to infix form in some respects. In Chapter 10, we saw that algebraic expressions with more than one operation didn't need parentheses if they were in prefix or postfix form. However, due to our familiarity with infix form, we will use it throughout most of the remainder of this book. Some operations, such as negation of numbers and complementation of sets, are not binary, but unary operators. Definition: Unary Operation. Let S be a nonempty set. A unary operator on S is a rule that assigns to each element of S a unique element of S. In other words, a unary operator is a function from S into S. COMMON PROPERTIES OF OPERATIONS Whenever an operation on a set is encountered, there are several properties that should immediately come to mind. To effectively make use of an operation, you should know which of these properties it has. By now, you should be familiar with most of these properties. We will list the most common ones here to refresh your memory and define them for the first time in a general setting. Let S be any set and * a binary operation on S. Properties that apply to a single binary operation: Let * be a binary operation on a set S * is commutative if a * b = b * a for all a, b œ S. * is associative if Ha * bL * c = a * Hb * cL for all a, b, c œ S. * has an identity if there exists an element, e, in S such that a * e = e * a = a for all a œ S. * has the inverse property if for each a œ S, there exists b œ S such that a*b = b*a = e. We call b an inverse of a. * is idempotent if a * a = a for all a œ S. Properties that apply to two binary operations: Let ù be a second binary operation on S. ù is left distributive over * if a ù Hb * cL = Ha ù bL * Ha ù cL for all a, b, c œ S. ù is right distributive over * if Hb * cLù a = Hbù aL * Hc ù aL for all a, b, c œ S. ù is distributive over * if ù is both left and right distributive over *. Let - be a unary operation. A unary operation — on S has the involution property if -H-aL = a for all a œ S. Finally, a property of sets, as they relate to operations. If T is a subset of S, we say that T is closed under * if a, b œ T implies that a * b œ T. In other words, by operating on elements of T with *, you can't obtain new elements that are outside of T. Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommerical-ShareAlike 3.0 United States License. Chapter 11 - Algebraic Systems Example 11.1.2. (a) The odd integers are closed under multiplication, but not under addition. (b) Let p be a proposition over U and let A be the set of propositions over U that imply p. That is; q œ A if q p. Then A is closed under both conjunction and disjunction. (c) The set positive integers that are multiples of 5 is closed under both addition and multiplication. Note: It is important to realize that the properties listed above depend on both the set and the operation(s). OPERATION TABLES If the set on which an operation is defined is small, a table is often a good way of describing the operation. For example, we might want to define Å⊕ on 80, 1, 2< by a + b if a + b < 3 a Å⊕b = : a + b - 3 if a + b ¥ 3 The table for Å⊕ is " Å⊕ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 The top row and left column are the column and row headings, respectively. To determine aÅ⊕b, find the entry in Row a and Column b. The following operation table serves to define * on 8i, j, k<. " * i j k i i i i j j j j k k k k Note that; j*k = j, yet k * j = k. Thus, * is not commutative. Commutivity is easy to identify in a table: the table must be symmetric with respect to the diagonal going from the top left to lower right. EXERCISES FOR SECTION 11.1 A Exercises 1. Determine the properties that the following operations have on the positive integers. (a) addition (b) multiplication (c) M defined by a M b = larger of a and b (d) m defined by a m b = smaller of a and b (e) @ defined by a ü b = ab 2. Which pairs of operations in Exercise 1 are distributive over one another? 3. Let * be an operation on a set S and A, B Œ S. Prove that if A and B are both closed under *, then A › B is also closed under *, but A ‹ B need not be. 4. How can you pick out the identity of an operation from its table? 5. Define a * b by a - b , the absolute value of a - b. Which properties does * have on the set of natural numbers, N? Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommerical-ShareAlike 3.0 United States License. Chapter 11 - Algebraic Systems 11.2 Algebraic Systems An algebraic system is a mathematical system consisting of a set called the domain and one or more operations on the domain. If V is the domain and *1 , *2 , …, *n are the operations, @V;*1, *2 , …, *nD denotes the mathematical system. If the context is clear, this notation is abbreviated to V. Example 11.2.1. (a) Let B* be the set of all finite strings of 0's and 1's including the null (or empty) string, l. An algebraic system is obtained by adding the operation of concatenation. The concatenation of two strings is simply the linking of the two strings together in the order indicated.